Math Telepathy

This is a stunning magic performed by the computer.

Follow the instructions. In the process you will be requested to compute the difference of two natural numbers (although on second thought, the omission of the decimal point appears not at all important.) See that you do not make a silly mistake. The magic fails in the face of ill preparedness. (As is well known in the magic circles, if a spell may go wrong, it will.) So, if you wish, you can use a calculator to compute that difference. In any event, it may be a good idea to double check your result.

Math Telepathy

After reshuffling a particular digit d may or may not change its position. In the decimal representation of the chosen number, d was a coefficient of a power of 10, say 10k. After reshuffling, the power may be different: 10m. Now, if we take the difference of two numbers term by term, i.e. seeking to subtract two terms related to the same digit even if its position changed, we'll get the sum of differences, like d(10k - 10m). Note the any power of 10 equals 1 modulo 9. This implies that every term d(10k - 10m) equals 0 modulo 9. Therefore, the whole difference of the two numbers (the chosen and the reshuffled) equals 0 modulo 9. When you communicate the computer all the digits, except the selected one, the computer finds the sum modulo 9 of the presented digits and subtracts the result from 9. This is exactly the digit you have omitted.

(Instead of subtracting the smaller number, one can subtract their digits in turn, or their sum just once. The result is a multiple of 9 and the trick works as before. In case of 2-digit numbers the triack has a different computer incarnation.)